Re: More strange behavior by ComplexExpand

*To*: mathgroup at smc.vnet.net*Subject*: [mg60622] Re: [mg60603] More strange behavior by ComplexExpand*From*: Pratik Desai <pdesai1 at umbc.edu>*Date*: Thu, 22 Sep 2005 02:08:15 -0400 (EDT)*References*: <200509210720.DAA08138@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Raul Martinez wrote: >To Mathgroup, > >I use Mathematica 5.2 with Mac OS X (Tiger). > >Add the following to a recent thread on the sometimes strange >behavior of ComplexExpand. > >I used ComplexExpand with an argument in which all the variables in >the argument of the function are real. Since ComplexExpand is >supposed to assume that all variables are real by default, one would >expect ComplexExpand to return the expression without change, but it >doesn't. > This is not exactly true, in mathematica all the variables are assumed complex (alteast what I have experienced so far) In my opinion not only doyou have to specifically assign your variable to be real by using Assuming or $Assumptions +Simplify, but also you have specifiy the values of "a" on the real line and hence your expression will change based on the complex expand algorithm Remove["Global`*"] $Assumptions = {-â?? < a < 0, t Ïµ Reals} s1=ComplexExpand[(a/Ï?)^(1/4) *Exp[-(a t^2)/2]] // Simplify >>((1 + I)*(a^2)^(1/8)*Sqrt[E^(a*t^2)])/(Sqrt[2]*E^(a*t^2)*Pi^(1/4)) $Assumptions = {0 < a < â??, t Ïµ Reals} s2=ComplexExpand[(a/Ï?)^(1/4) *Exp[-(a t^2)/2]] // Simplify >>(a^(1/4)*Sqrt[E^(a*t^2)])/(E^(a*t^2)*Pi^(1/4)) $Assumptions = {a Ïµ Reals, t Ïµ Reals} s3=ComplexExpand[(a/Ï?)^(1/4) *Exp[-(a t^2)/2]] // Simplify >>((a^2)^(1/8)*Sqrt[E^(a*t^2)]*(Cos[Arg[a]/4] + I*Sin[Arg[a]/4]))/(E^(a*t^2)*Pi^(1/4)) s4=Simplify[(a/Ï?)^(1/4) *Exp[-(a t^2)/2]] >>a^(1/4)/(E^((a*t^2)/2)*Pi^(1/4)) fundamentally you are asking by, using complexexpand, to expand your given function in a complex way.... >Instead, here is what it does: > >In[1]:= > > ComplexExpand[ (a / Pi)^(1/4) Exp[ (-(a t^2)/2 ] ] > >Out[2]:= > > (Exp[-a t^2] Sqrt[Exp[a t^2]] (a^2)^(1/8) Cos[Arg[a] / 4]) / Pi^ >(1/4) + i (Exp[-a t^2] Sqrt[Exp[a t^2]] (a^2)^(1/8) Sin[Arg[a] / >4]) / Pi^(1/4). > >I have inserted parentheses in a few places to improve the legibility >of the expressions. > >ComplexExpand treats the variable "a" as complex, but "t" as real. >This is puzzling to say the least. Moreover, it renders a^(1/4) as >(a^2)^(1/8), which seems bizarre. > > If you look closely at your expression, the only issue of complexity occurs with your "a" variable because appears as a radical which may have complex nature based on where it is defined on the Real number line >My interest is not in obtaining the correct result, which is easy to >do. Rather, I bring this up as yet another example of the >unreliability of ComplexExpand. In case anyone is wondering why I >would use ComplexExpand on an expression I know to be real, the >reason is that the expression in question is a factor in a larger >expression that contains complex variables. Applied to the larger >expression, ComplexExpand returned an obviously incorrect expansion >that I traced to the treatment of the example shown above. > >I welcome comments and suggestions. > >Thanks in advance, > >Raul Martinez > > > > > > > > Hope this helps Pratik Desai -- Pratik Desai Graduate Student UMBC Department of Mechanical Engineering Phone: 410 455 8134

**References**:**More strange behavior by ComplexExpand***From:*Raul Martinez <raulm231@comcast.net>